Mesh Analysis

Write KVL around every mesh using mesh currents as unknowns.

Easy

Assign a clockwise mesh current to each window of a planar circuit.
Apply KVL around each mesh; a shared resistor carries the difference of the two mesh currents.
Solve.

For two meshes sharing $R_3$ :

$V_1 = I_1(R_1 + R_3) - I_2 R_3 \qquad 0 = -I_1 R_3 + I_2(R_2 + R_3)$

A current source on a shared branch creates a supermesh. Choose mesh analysis when the circuit has fewer meshes than nodes, or many voltage sources.

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